Intro
Issue
TROLL model currently compute leaf lifespan with Reich’s allometry (Reich et al. 1991). But we have shown that Reich’s allometry is underestimating leaf lifespan for low LMA species. Moreover simulations estimated unrealistically low aboveground biomass for low LMA species. We assumed Reich’s allometry underestimation of leaf lifespan for low LMA species being the source of unrealistically low aboveground biomass inside TROLL simulations. We decided to find a better allometry with Wright et al. (2004) GLOPNET dataset.
We tested different models starting from complet model Mcomp: \[ {LL_s}_j \sim \mathcal{logN}({\beta_0}*e^{{\beta_1}_s*{LMA_s}_j^{{\beta_3}_s} - {\beta_2}_s*{Nmass_s}_j^{{\beta_4}_s}},\,\sigma)\,\]
\[s=1,...,S_{=4}~, ~~j=1,...,n_s\] \[{\beta_i}_s \sim \mathcal{N}({\beta_i},\,\sigma_i)\,^I, ~~(\beta_i, \sigma, \sigma_i) \sim \mathcal{\Gamma}(0.001,\,0.001)\,^{2I+1}\] We tested models M1 to M9 detailed in following tabs to find the better trade-off between:
- Complexity (and number of parameters)
- Convergence
- Likelihood
- Prediction quality with Root Mean Square Error of Prediction (RMSEP)
RMSEP was computed by fitting a model without a dataset and evaluating it with the same dataset. RMSEP in results are mean RMSEP for each dataset. Results are shown for each models in eahc model tabs and summarized in Results tab.
LL graph
Figure 1: Leaf mass per area (LMA), leaf nitrogen content (Nmass) and leaflifespan (LL). Leaf mass per area (LMA in \(g.m^{-2}\)), leaf nitrogen content (Nmass, in \(mg.g^-1\)) and leaf lifespan (LL in \(months\)) are taken in GLOPNET dataset from Wright et al. (2004).
M1
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - {\beta_2}_s*N,\sigma)\,\] Maximum likekihood of 9.0158557 and RMSEP of 13.207
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M2
Model
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -2.1040528 and RMSEP of 14.361
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M3
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 16.5819461 and RMSEP of 291.664
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M4
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - N,\sigma)\,\] Maximum likekihood of 1.8743912 and RMSEP of 12.767
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M5
Model
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -0.7373547 and RMSEP of 14.215
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M6
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 5.457475 and RMSEP of 12.369
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M7
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA,\sigma)\,\] Maximum likekihood of 4.2155493 and RMSEP of 14.391
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M8
Model
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of -5.4453681 and RMSEP of 14.117
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M9
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of 9.3372588 and RMSEP of 16.06
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
Results
Column
Model M1 seemed to have shown the best trade-off between complexity (\(K=6\) parameters), convergence (see tab M1), likelihood, and prediction quality (see table 1). Figure 2 presents model prediction confidence interval and figure 3 compare Reich’s allometry and model M1 predictions with species functional traits used in TROLL. Nevertheless we will test other predictors gathered from TRY database following a model with the same response curve.
Table 1: Models likelihood and prediction quality.Root mean square errof of prediction (RMSEP) was computed by fitting a model without a dataset and evaluating it with the same dataset. RMSEP in results are mean RMSEP for each dataset.
| ML | RMSEP | |
|---|---|---|
| M1 | 9.016 | 13.20692 |
| M2 | -2.104 | 14.36052 |
| M3 | 16.582 | 291.66404 |
| M4 | 1.874 | 12.76683 |
| M5 | -0.737 | 14.21515 |
| M6 | 5.457 | 12.36851 |
| M7 | 4.216 | 14.39148 |
| M8 | -5.445 | 14.11743 |
| M9 | 9.337 | 16.06047 |
\[LL = 14.277*e^{0.007 *LMA -0.411*Nmass }\]
Column
Figure 2: Leaflifespan predictions for model M1 with leaf mass per area (LMA), and leaf nitrogen content (Nmass). Leaf lifespan (LL in \(months\)) is predicted with model M1 fit. Leaf mass per area (LMA in \(g.m^{-2}\)) and leaf nitrogen content (Nmass, in %) are taken in GLOPNET dataset from Wright et al. (2004).
Figure 3: Leaflifespan predictions for model M1 and Reich’s allometry with leaf mass per area (LMA), and leaf nitrogen content (Nmass). Leaf lifespan (LL in \(months\)) is predicted with model M1 fit or Reich’s allometry (Reich et al. 1991). Leaf mass per area (LMA in \(g.m^{-2}\)) and leaf nitrogen content (Nmass, in %) are taken in BRIDGE dataset used by TROLL (Maréchaux & Chave).
References
Maréchaux, I. & Chave, J. Joint simulation of carbon and tree diversity in an Amazonian forest with an individual-based forest model. Inprep, 1–13.
Reich, P.B., Uhl, C., Walters, M.B. & Ellsworth, D.S. (1991). Leaf lifespan as a determinant of leaf structure and function among 23 amazonian tree species. Oecologia, 86, 16–24.
Wright, I.J., Reich, P.B., Westoby, M., Ackerly, D.D., Baruch, Z., Bongers, F., Cavender-Bares, J., Chapin, T., Cornelissen, J.H.C., Diemer, M. & Others. (2004). The worldwide leaf economics spectrum. Nature, 428, 821–827.